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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 21

Pack n equal circles into the true regular m-gon with centre (1,1), circumradius 1 and first vertex (2,1), maximising their radius.

Instancem = 5 · n = 21
ObjectiveMaximize circle radius
Best known, unproven0.165404018640Source-precision interval: 0.165404018639–0.165404018641 (values inside match)exhibited site certificate 0.165404018Packomania (compiled by Eckard Specht), m=5, n=21. Paolo Amore, Circle packing in regular polygons , Phys. Fluids 35 , 027130 (2023). Coordinates imported from the public archive, normalized, decimalized and exactly verified by MinMax Arena. The web-table target and grid certificate are separate.

Formal definition

  • ContainerThe editor and certificate normalize to the true regular m-gon with centre (1,1), circumradius 1 and first vertex (2,1). This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.
  • ConstraintsExactly n congruent shapes lie wholly in the container with pairwise disjoint interiors. Polygon rotations are independent, not limited to quarter turns.
  • PrecisionInput decimals are exact rationals; regular-polygon vertices are algebraic, not rounded templates. Feasibility has no floating-point tolerance. A finite-decimal certificate is not a continuous optimality proof.
  • ObjectiveCircle radius r, larger is better; the container circumradius is 1.
Open the full editor ↗
VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Source and certificate

External targets are converted with their printed precision; decimal coordinate certificates are separate. Reconstruction loss never lowers the literature target, and a source construction is not automatically an optimality proof.

Source
The record-holding arrangement for Equal circles in regular polygons m = 5 · n = 21, 0.165404018
Current leader

0.165404018

circle radius

Answer sourcePaolo Amore
Solution methodPublished reference construction
Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

ANSWER FORMAT

How to write your answer

the true regular m-gon with centre (1,1), circumradius 1 and first vertex (2,1)

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.356387024",
      "y": "1.467612198"
    },
    {
      "turn": "0",
      "x": "0.356387024",
      "y": "0.863195839"
    },
    {
      "turn": "0",
      "x": "0.356387024",
      "y": "0.532387802"
    },
    {
      "turn": "0",
      "x": "0.490938775",
      "y": "1.165404019"
    },
    {
      "turn": "0",
      "x": "0.671004164",
      "y": "1.569837504"
    },
    {
      "turn": "0",
      "x": "0.685382861",
      "y": "0.897774695"
    },
    {
      "turn": "0",
      "x": "0.685382861",
      "y": "0.566966657"
    },
    {
      "turn": "0",
      "x": "0.805555914",
      "y": "1.267629324"
    },
    {
      "turn": "0",
      "x": "0.931221142",
      "y": "0.345612875"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.535258648"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "0.669191963"
    },
    {
      "turn": "0",
      "x": "1.194444086",
      "y": "1.267629324"
    },
    {
      "turn": "0",
      "x": "1.245838281",
      "y": "1.756612431"
    },
    {
      "turn": "0",
      "x": "1.245838281",
      "y": "0.243387569"
    },
    {
      "turn": "0",
      "x": "1.314617139",
      "y": "0.897774695"
    },
    {
      "turn": "0",
      "x": "1.314617139",
      "y": "0.566966657"
    },
    {
      "turn": "0",
      "x": "1.440282367",
      "y": "1.488983107"
    },
    {
      "turn": "0",
      "x": "1.509061225",
      "y": "1.165404019"
    },
    {
      "turn": "0",
      "x": "1.601105304",
      "y": "0.732370676"
    },
    {
      "turn": "0",
      "x": "1.795549389",
      "y": "1.000000000"
    }
  ],
  "radius": "0.165404018"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 21,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.356387024",
      "y": "1.467612198"
    },
    {
      "turn": "0",
      "x": "0.356387024",
      "y": "0.863195839"
    },
    {
      "turn": "0",
      "x": "0.356387024",
      "y": "0.532387802"
    },
    {
      "turn": "0",
      "x": "0.490938775",
      "y": "1.165404019"
    },
    {
      "turn": "0",
      "x": "0.671004164",
      "y": "1.569837504"
    },
    {
      "turn": "0",
      "x": "0.685382861",
      "y": "0.897774695"
    },
    {
      "turn": "0",
      "x": "0.685382861",
      "y": "0.566966657"
    },
    {
      "turn": "0",
      "x": "0.805555914",
      "y": "1.267629324"
    },
    {
      "turn": "0",
      "x": "0.931221142",
      "y": "0.345612875"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.535258648"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "0.669191963"
    },
    {
      "turn": "0",
      "x": "1.194444086",
      "y": "1.267629324"
    },
    {
      "turn": "0",
      "x": "1.245838281",
      "y": "1.756612431"
    },
    {
      "turn": "0",
      "x": "1.245838281",
      "y": "0.243387569"
    },
    {
      "turn": "0",
      "x": "1.314617139",
      "y": "0.897774695"
    },
    {
      "turn": "0",
      "x": "1.314617139",
      "y": "0.566966657"
    },
    {
      "turn": "0",
      "x": "1.440282367",
      "y": "1.488983107"
    },
    {
      "turn": "0",
      "x": "1.509061225",
      "y": "1.165404019"
    },
    {
      "turn": "0",
      "x": "1.601105304",
      "y": "0.732370676"
    },
    {
      "turn": "0",
      "x": "1.795549389",
      "y": "1.000000000"
    }
  ],
  "radius": "0.165404018"
}

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4. · Verifier v1.0.0

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